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(1)

“best matching” et d´elais

Jean Dolbeault

http://www.ceremade.dauphine.fr/ ∼ dolbeaul Ceremade, Universit´e Paris-Dauphine

12 Mai 2015

(2)

A. Fast diffusion equations: entropy, linearization, inequalities,

improvements

1

entropy methods

2

linearization of the entropy

3

improved Gagliardo-Nirenberg inequalities

(3)

Fast diffusion equations: entropy

methods

(4)

Existence, classical results

u t = ∆u m x ∈ R d , t > 0

Self-similar (Barenblatt) function: U (t) = O(t −d/(2 −d(1 −m)) ) as t → + ∞

[Friedmann, Kamin, 1980] k u(t, · ) − U (t, · ) k L

= o(t −d/(2 −d(1 −m)) )

d − 1 d

m fast diffusion equation

porous media equation heat equation

1

d − 2 d

global existence in L 1 extinction in finite time

Existence theory, critical values of the parameter m

(5)

Time-dependent rescaling, Free energy

Time-dependent rescaling: Take u(τ, y) = R −d (τ) v (t, y /R(τ)) where

dR

dτ = R d(1 −m) 1 , R(0) = 1 , t = log R The function v solves a Fokker-Planck type equation

∂v

∂t = ∆v m + ∇ · (x v) , v | τ=0 = u 0

[Ralston, Newman, 1984] Lyapunov functional:

Generalized entropy or Free energy F [v ] :=

Z

R

d

v m m − 1 + 1

2 | x | 2 v

dx − F 0

Entropy production is measured by the Generalized Fisher information

d

dt F [v ] = −I [v ] , I [v ] :=

Z

R

d

v ∇ v m

v + x

2

dx

(6)

Relative entropy and entropy production

Stationary solution: choose C such that k v k L

1

= k u k L

1

= M > 0 v (x ) := C + 1 2m −m | x | 2 − 1/(1 −m)

+

Relative entropy: Fix F 0 so that F [v ] = 0 Entropy – entropy production inequality

Theorem

d ≥ 3, m ∈ [ d d 1 , + ∞ ), m > 1 2 , m 6 = 1 I [v] ≥ 2 F [v ]

Corollary

A solution v with initial data u 0 ∈ L 1 + ( R d ) such that | x | 2 u 0 ∈ L 1 ( R d ),

u 0 m ∈ L 1 ( R d ) satisfies F [v(t , · )] ≤ F [u 0 ] e 2 t

(7)

An equivalent formulation: Gagliardo-Nirenberg inequalities

F [v] = Z

R

d

v m m − 1 + 1

2 | x | 2 v

dx −F 0 ≤ 1 2 Z

R

d

v ∇ v m

v + x

2

dx = 1 2 I [v]

Rewrite it with p = 2m− 1 1 , v = w 2p , v m = w p+1 as 1

2 2m

2m − 1 2 Z

R

d

|∇ w | 2 dx + 1

1 − m − d Z

R

d

| w | 1+p dx − K ≥ 0 for some γ, K = K 0 R

R

d

v dx = R

R

d

w 2p dx γ

w = w = v 1/2p is optimal Theorem

[Del Pino, J.D.] With 1 < p ≤ d d 2 (fast diffusion case) and d ≥ 3 k w k L

2p

( R

d

) ≤ C p,d GN k∇ w k θ L

2

( R

d

) k w k 1 L

p+1

θ ( R

d

)

C p,d GN =

y(p− 1)

2

2πd

θ2

2y − d 2y

2p1

Γ(y)

Γ(y −

d2

)

θd

, θ = p(d+2 d(p− (d 1) 2)p) , y = p+1 p 1

(8)

... a proof by the Bakry-Emery method

Consider the generalized Fisher information I [v ] :=

Z

R

d

v | z | 2 dx with z := ∇ v m v + x and compute

d

dt I [v (t, · )]+2 I [v (t , · )] = − 2 (m − 1) Z

R

d

u m (divz ) 2 dx − 2 X d i, j=1

Z

R

d

u m (∂ i z j ) 2 dx

the Fisher information decays exponentially:

I [v(t, · )] ≤ I [u 0 ] e 2t lim t →∞ I [v(t, · )] = 0 and lim t →∞ F [v (t , · )] = 0

d dt

I [v (t, · )] − 2 F [v(t , · )]

≤ 0 means I [v ] ≥ 2 F [v]

[Carrillo, Toscani], [Juengel, Markowich, Toscani], [Carrillo, Juengel,

Markowich, Toscani, Unterreiter], [Carrillo, V´ azquez]

(9)

The Bakry-Emery method: details (1/2)

With z (x, t) := η ∇ u m− 1 − 2 x, the equation can be rewritten as

∂u

∂t + ∇ · (u z ) = 0

(up to a time rescaling, which introduces a factor 2) and we have

∂z

∂t = η (1 − m) ∇ u m− 2 ∇ · (u z)

and ∇ ⊗ z = η ∇ ⊗ ∇ u m− 1 − 2 Id d

dt Z

R

d

u | z | 2 dx = Z

R

d

∂u

∂t | z | 2 dx

| {z }

( I )

+ 2 Z

R

d

u z · ∂z

∂t dx

| {z }

( II )

(I) = Z

R

d

∂u

∂t | z | 2 dx = Z

R

d

∇ · (u z ) | z | 2 dx

= 2 η (1 − m) Z

R

d

u m− 2 ( ∇ u · z ) 2 dx + 2 η (1 − m) Z

R

d

u m− 1 ( ∇ u · z ) ( ∇ · z) dx + 2 η (1 − m)

Z

R

d

u m− 1 (z ⊗ ∇ u) : ( ∇ ⊗ z ) dx − 4 Z

R

d

u | z | 2 dx

(10)

The Bakry-Emery method: details (2/2)

(II) = 2 Z

R

d

u z · ∂z

∂t dx

= − 2 η (1 − m) Z

R

d

u m ( ∇ · z) 2 + 2 u m 1 ( ∇ u · z ) ( ∇ · z ) + u m 2 ( ∇ u · z ) 2 dx Z

R

d

∂u

∂t | z | 2 dx + 4 Z

R

d

u | z | 2 dx

= − 2 η (1 − m) Z

R

d

u m− 2

u 2 ( ∇ · z) 2 + u ( ∇ u · z ) ( ∇ · z )

= − 2 η 1 − m m

Z

R

d

u m

|∇ z | 2 − (1 − m) ( ∇ · z ) 2 dx By the arithmetic geometric inequality, we know that

|∇ z | 2 − (1 − m) ( ∇ · z ) 2 ≥ 0

if 1 − m ≤ 1/d, that is, if m ≥ m 1 = 1 − 1/d

(11)

Fast diffusion: finite mass regime

Inequalities...

d − 1 d

m 1

d − 2 d

global existence in L 1

Bakry-Emery method (relative entropy) v m ∈ L 1 , x

2

∈ L 1 Sobolev

Gagliardo-Nirenberg logarithmic Sobolev

d d +2

v

... existence of solutions of u t = ∆u m

(12)

Fast diffusion equations: the infinite mass regime by

linearization of the entropy

(13)

Extension to the infinite mass regime, finite time vanishing

If m > m c := d d 2 ≤ m < m 1 , solutions globally exist in L 1 ( R d ) and the Barenblatt self-similar solution has finite mass

For m ≤ m c , the Barenblatt self-similar solution has infinite mass Extension to m ≤ m c ? Work in relative variables !

d−1 d

m 1

d−2 d

glob al existenc e i n L

1

Bakry-Emer y m etho d ( relati ve en trop y) v

m

∈ L

1

, x

2

∈ L

1

d d+2

v v

0

, V

D

∈ L

1

v

0

− V

D

∈ L

1

V

D1

− V

D0

∈ L

1

[ V

D1

V

D0

] < ∞ [ V

D1

V

D0

] = ∞

m

1 d−4

d−2

V

D1

− V

D0

6∈ L

1

m

c

m

Gagliardo-Nire nb er g F

F

(14)

Entropy methods and linearization: intermediate asymptotics, vanishing

[A. Blanchet, M. Bonforte, J.D., G. Grillo, J.L. V´ azquez]

∂u

∂τ = −∇ · (u ∇ u m− 1 ) = 1 − m

m ∆u m (1)

m c < m < 1, T = + ∞ : intermediate asymptotics, τ → + ∞ R(τ) := (T + τ)

d(m1mc)

0 < m < m c , T < + ∞ : vanishing in finite time lim τ րT u(τ, y) = 0 R(τ) := (T − τ)

d(mc1m)

Self-similar Barenblatt type solutions exists for any m t := 1 2 m log

R (τ) R(0)

and x := q

1 2 d | m − m

c

|

y

R(τ)

Generalized Barenblatt profiles: V D (x) := D + | x | 2

m11

(15)

Sharp rates of convergence

Assumptions on the initial datum v 0

(H1) V D

0

≤ v 0 ≤ V D

1

for some D 0 > D 1 > 0

(H2) if d ≥ 3 and m ≤ m , (v 0 − V D ) is integrable for a suitable D ∈ [D 1 , D 0 ]

Theorem

[Blanchet, Bonforte, J.D., Grillo, V´ azquez] Under Assumptions (H1)-(H2), if m < 1 and m 6 = m := d− d− 4 2 , the entropy decays according to

F [v (t, · )] ≤ C e 2 (1 m) Λ

α,d

t ∀ t ≥ 0

where Λ α,d > 0 is the best constant in the Hardy–Poincar´e inequality Λ α,d

Z

R

d

| f | 2 dµ α − 1 ≤ Z

R

d

|∇ f | 2 dµ α ∀ f ∈ H 1 (d µ α )

with α := 1/(m − 1) < 0, dµ α := h α dx, h α (x) := (1 + | x | 2 ) α

(16)

Plots (d = 5)

λ01=−4α−2d

λ10=−2α λ11=−6α−2 (d+ 2) λ02=−8α−4 (d+ 2)

λ20=−4α λ30 λ21λ

12 λ03

λcontα,d:=14(d+ 2α−2)2

α=−d α=−(d+ 2)

α=−d+2 2

α=−d−22 α=−d+62

α 0 Essential spectrum ofLα,d

α=−d−1d2 α=−d−1d+42 α=− d+22

2d (d= 5) Spectrum ofLα,d

mc=d−2d

m1=d−1d m2=d+1d+2

e m1=d+2d

e m2=d+4d+6

m Spectrum of (1−m)L1/(m−1),d

(d= 5)

Essential spectrum of (1

1

m)L1/(m−1),d

2 4 6

(17)

Improved asymptotic rates

[Bonforte, J.D., Grillo, V´ azquez] Assume that m ∈ (m 1 , 1), d ≥ 3.

Under Assumption (H1), if v is a solution of the fast diffusion equation with initial datum v 0 such that R

R

d

x v 0 dx = 0, then the asymptotic convergence holds with an improved rate corresponding to the improved spectral gap.

m1=d+2d

m1=d−1 d

m2=d+4d+6 m2=d+1d+2

4

2

m 1 mc=d−2

d (d= 5)

γ(m)

0

(18)

Higher order matching asymptotics

[J.D., G. Toscani] For some m ∈ (m c , 1) with m c := (d − 2)/d, we consider on R d the fast diffusion equation

∂u

∂τ + ∇ · u ∇ u m 1

= 0

Without choosing R, we may define the function v such that u(τ, y + x 0 ) = R d v (t, x) , R = R(τ) , t = 1 2 log R , x = y

R Then v has to be a solution of

∂v

∂t + ∇ · h v

σ

d2

(m−m

c

) ∇ v m− 1 − 2 x i

= 0 t > 0 , x ∈ R d with (as long as we make no assumption on R)

2 σ

d2

(m−m

c

) = R 1 −d (1 −m) dR

(19)

Refined relative entropy

Consider the family of the Barenblatt profiles B σ (x) := σ

d2

C M + 1 σ | x | 2

m11

∀ x ∈ R d (2) Note that σ is a function of t : as long as dt 6 = 0, the Barenblatt profile B σ is not a solution (it plays the role of a local Gibbs state) but we may still consider the relative entropy

F σ [v] := 1 m − 1

Z

R

d

v m − B σ m − m B σ m 1 (v − B σ ) dx The time derivative of this relative entropy is

d

dt F σ(t) [v(t , · )] = dσ dt

d dσ F σ [v]

| σ=σ(t)

| {z }

choose it = 0

⇐⇒ Minimize F σ [v ] w.r.t. σ ⇐⇒ R

R

d

| x | 2 B σ dx = R

R

d

| x | 2 v dx

+ m

m − 1 Z

R

d

v m 1 − B σ(t) m− 1 ∂v

∂t dx

(20)

The entropy / entropy production estimate

Using the new change of variables, we know that d

dt F σ(t) [v (t , · )] = − m σ(t)

d2

(m−m

c

) 1 − m

Z

R

d

v ∇ h

v m 1 − B σ(t) m− 1 i

2

dx

Let w := v/B σ and observe that the relative entropy can be written as F σ [v] = m

1 − m Z

R

d

h w − 1 − 1

m w m − 1 i B σ m dx (Repeating) define the relative Fisher information by

I σ [v] :=

Z

R

d

1

m − 1 ∇

(w m− 1 − 1) B σ m− 1

2

B σ w dx

so that d

dt F σ(t) [v (t , · )] = − m (1 − m) σ(t ) I σ(t) [v (t , · )] ∀ t > 0

When linearizing, one more mode is killed and σ(t) scales out

(21)

Improved rates of convergence

Theorem (J.D., G. Toscani)

Let m ∈ ( m e 1 , 1), d ≥ 2, v 0 ∈ L 1 + ( R d ) such that v 0 m , | y | 2 v 0 ∈ L 1 ( R d ) F [v(t, · )] ≤ C e 2 γ(m) t ∀ t ≥ 0

where

γ(m) =

 

 

 

((d − 2) m − (d − 4))

2

4 (1 −m) if m ∈ ( m e 1 , m e 2 ] 4 (d + 2) m − 4 d if m ∈ [ m e 2 , m 2 ]

4 if m ∈ [m 2 , 1)

(22)

Spectral gaps and best constants

m

c

=

d−2d

0

m

1

=

d−1d

m

2

=

d+1d+2

e m

2

:=

d+4d+6

m 1

2 4

Case 1 Case 2 Case 3

γ(m)

(d= 5)

e

m

1

:=

d+2d

(23)

Comments

1

A result by [Denzler, Koch, McCann]Higher order time asymptotics of fast diffusion in Euclidean space: a dynamical systems approach

2

The constant C in

F [v(t, · )] ≤ C e 2 γ(m) t ∀ t ≥ 0

can be made explicit, under additional restrictions on the initial

data [Bonforte, J.D., Grillo, V´ azquez]

(24)

An explicit constant C ?

d

dt F [w (t, · )] = − I [w (t , · )] ∀ t > 0 h m 2

Z

R

d

| f | 2 V D 2 m dx ≤ 2 F [w ] ≤ h 2 m Z

R

d

| f | 2 V D 2 m dx where f := (w − 1) V D m 1 , h := max { sup R

d

w (t, · ), 1/inf R

d

w (t , · ) } Z

R

d

|∇ f | 2 V D dx ≤ h 5 2m I [w ] + d (1 − m)

h 4(2 m) − 1 Z

R

d

| f | 2 V D 2 −m dx 0 ≤ h − 1 ≤ C F

1

−m d+2−(d+1)m

Corollary

F [w (t, · )] ≤ G t, h(0), F [w (0, · )]

for any t ≥ 0, where dG

dt = − 2 Λ α,d − Y (h)

[1 + X(h)] h 2 m G , h = 1 + C G

d+21−(d+1)mm

, G (0) = F [w (0, · )]

(25)

Gagliardo-Nirenberg and Sobolev inequalities :

improvements

[J.D., G. Toscani]

(26)

Best matching Barenblatt profiles

(Repeating) Consider the fast diffusion equation

∂u

∂t + ∇ · h u

σ

d2

(m m

c

) ∇ u m 1 − 2 x i

= 0 t > 0 , x ∈ R d with a nonlocal, time-dependent diffusion coefficient

σ(t ) = 1 K M

Z

R

d

| x | 2 u(x, t) dx , K M :=

Z

R

d

| x | 2 B 1 (x) dx where

B λ (x) := λ

d2

C M + λ 1 | x | 2

m−11

∀ x ∈ R d and define the relative entropy

F λ [u] := 1 m − 1

Z

R

d

u m − B λ m − m B λ m 1 (u − B λ )

dx

(27)

Three ingredients for global improvements

1

inf λ>0 F λ [u(x , t )] = F σ(t) [u(x, t)] so that d

dt F σ(t) [u(x, t)] = −J σ(t) [u( · , t)]

where the relative Fisher information is J λ [u] := λ

d2

(m m

c

) m

1 − m Z

R

d

u ∇ u m 1 − ∇ B λ m− 1 2 dx

2

In the Bakry-Emery method, there is an additional (good) term 4

"

1 + 2 C m,d F σ(t) [u( · , t )]

M γ σ 0

d2

(1 −m)

# d

dt F σ(t) [u( · , t )]

≥ d

dt J σ(t) [u( · , t)]

3

The Csisz´ ar-Kullback inequality is also improved F σ [u] ≥ m

8 R

R

d

B 1 m dx C M 2 k u − B σ k 2 L

1

( S

d

)

(28)

improved decay for the relative entropy

0.0 0.2 0.4 0.6 0.8 1.0

0.2 0.4 0.6 0.8 1.0

t 7→ f(t) e

4t

f (0)

t (a)

(b) (c)

Figure: Upper bounds on the decay of the relative entropy: t 7→ f (t) e

4t

/f (0) (a): estimate given by the entropy-entropy production method

(b): exact solution of a simplified equation

(c): numerical solution (found by a shooting method)

(29)

A Csisz´ar-Kullback(-Pinsker) inequality

Let m ∈ ( m e 1 , 1) with m e 1 = d+2 d and consider the relative entropy F σ [u] := 1

m − 1 Z

R

d

u m − B σ m − m B σ m 1 (u − B σ ) dx

Theorem

Let d ≥ 1, m ∈ ( m e 1 , 1) and assume that u is a nonnegative function in L 1 ( R d ) such that u m and x 7→ | x | 2 u are both integrable on R d . If k u k L

1

( S

d

) = M and R

R

d

| x | 2 u dx = R

R

d

| x | 2 B σ dx, then F σ [u]

σ

d2

(1 m) ≥ m 8 R

R

d

B 1 m dx

C M k u − B σ k L

1

( S

d

) + 1 σ

Z

R

d

| x | 2 | u − B σ | dx

2

(30)

Csisz´ar-Kullback(-Pinsker): proof (1/2)

Let v := u/B σ and dµ σ := B σ m dx Z

R

d

(v − 1) dµ σ = Z

R

d

B σ m− 1 (u − B σ ) dx

= σ

d2

(1 −m) C M

Z

R

d

(u − B σ ) dx + σ

d2

(m

c

−m) Z

R

d

| x | 2 (u − B σ ) dx = 0 Z

R

d

(v − 1) dµ σ = Z

v>1

(v − 1) dµ σ − Z

v<1

(1 − v ) dµ σ = 0 Z

R

d

| v − 1 | dµ σ = Z

v>1

(v − 1) dµ σ + Z

v<1

(1 − v) dµ σ

Z

R

d

| u − B σ | B σ m− 1 dx = Z

R

d

| v − 1 | dµ σ = 2 Z

v<1 | v − 1 | dµ σ

(31)

Csisz´ar-Kullback(-Pinsker): proof (2/2)

A Taylor expansion shows that F σ [u] = 1

m − 1 Z

R

d

v m − 1 − m (v − 1)

d µ σ = m 2

Z

R

d

ξ m− 2 | v − 1 | 2 dµ σ

≥ m 2

Z

v<1 | v − 1 | 2 dµ σ

Using the Cauchy-Schwarz inequality, we get R

v<1 | v − 1 | dµ σ 2

= R

v<1 | v − 1 | B σ

m2

B σ

m2

dx 2

≤ R

v <1 | v − 1 | 2 dµ σ R

R

d

B σ m dx and finally obtain that

F σ [u] ≥ m 2

R

v<1 | v − 1 | dµ σ 2

R

R

d

B σ m dx = m 8

R

R

d

| u − B σ | B σ m 1 dx 2

R

R

d

B σ m dx

(32)

An improved Gagliardo-Nirenberg inequality: the setting

The inequality

k f k L

2p

( S

d

) ≤ C GN p,d k∇ f k θ L

2

( S

d

) k f k 1 L

p+1

θ ( S

d

)

with θ = θ(p) := p− p 1 d+2 d p (d 2) , 1 < p ≤ d d 2 if d ≥ 3 and

1 < p < ∞ if d = 2, can be rewritten, in a non-scale invariant form, as Z

R

d

|∇ f | 2 dx + Z

R

d

| f | p+1 dx ≥ K p,d

Z

R

d

| f | 2 p dx γ

with γ = γ(p, d ) := d+2 d −p p (d (d− 4) 2) . Optimal function are given by f M,y,σ (x ) = 1

σ

d2

C M + | x − y | 2 σ

p−11

∀ x ∈ R d where C M is determined by R

R

d

f M,y 2p dx = M M d :=

f M,y,σ : (M, y, σ) ∈ M d := (0, ∞ ) × R d × (0, ∞ )

(33)

An improved Gagliardo-Nirenberg inequality (1/2)

Relative entropy functional R (p) [f ] := inf

g ∈ M

(p)

d

Z

R

d

h g 1 −p | f | 2p − g 2p

− p+1 2 p | f | p+1 − g p+1 i dx

Theorem

Let d ≥ 2, p > 1 and assume that p < d /(d − 2) if d ≥ 3. If R

R

d

| x | 2 | f | 2 p dx R

R

d

| f | 2 p dx γ = d (p d+2 1) −p σ (d−

M

γ

2)

−1

, σ (p) :=

4 d+2 (p− 1) p

2

(d (p+1) 2)

d−p4(dp−4)

for any f ∈ L p+1 ∩ D 1,2 ( R d ), then we have Z

R

d

|∇ f | 2 dx+

Z

R

d

| f | p+1 dx − K p,d

Z

R

d

| f | 2 p dx γ

≥ C p,d R (p) [f ] 2 R

R

d

| f | 2p dx γ

(34)

An improved Gagliardo-Nirenberg inequality (2/2)

A Csisz´ ar-Kullback inequality

R (p) [f ] ≥ C CK k f k 2 L p

2p

( S

d

) 2) inf

g ∈ M

(p)

d

k| f | 2 p − g 2p k 2 L

1

( S

d

)

with C CK = p− p+1 1 d+2 32 p p (d 2) σ d

p−1 4p

∗ M 1 γ . Let C p,d := C d,p C CK 2

Corollary

Under previous assumptions, we have Z

R

d

|∇ f | 2 dx + Z

R

d

| f | p+1 dx − K p,d

Z

R

d

| f | 2p dx γ

≥ C p,d k f k 2p L

2p

( S

d

) 4) inf

g ∈ M

d

(p) k| f | 2 p − g 2 p k 4 L

1

( S

d

)

(35)

Conclusion 1: improved inequalities

We have found an improvement of an optimal Gagliardo-Nirenberg inequality, which provides an explicit measure of the distance to the manifold of optimal functions.

The method is based on the nonlinear flow

The explicit improvement gives (is equivalent to) an improved

entropy – entropy production inequality

(36)

Conclusion 2: improved rates

If m ∈ (m 1 , 1), with

f (t) := F σ(t) [u( · , t)]

σ(t ) = 1 K M

Z

R

d

| x | 2 u(x, t) dx j(t) := J σ(t) [u( · , t)]

J σ [u] := m σ

d2

(m m

c

) 1 − m

Z

R

d

u ∇ u m− 1 − ∇ B m− σ 1 2 dx we can write a system of coupled ODEs

 

 

 

f = − j ≤ 0 σ = − 2 d (1 m K m)

2

M

σ

d2

(m−m

c

) f ≤ 0 j + 4 j = d 2 (m − m c ) h

j

σ + 4 d (1 − m) σ f i σ − r

(3)

In the rescaled variables, we have found an improved decay (algebraic

rate) of the relative entropy. This is a new nonlinear effect, which

matters for the initial time layer

(37)

Conclusion 3: Best matching Barenblatt profiles are delayed

Let u be such that

v(τ, x) = µ d R(D τ) d u

1

2 log R(D τ), µ x R(D τ)

with τ 7→ R(τ) given as the solution to 1

R dR dτ =

µ 2 K M

Z

R

d

| x | 2 v (τ, x) dx

d2

(m−m

c

)

, R(0) = 1 Then

1 R

dR dτ =

R 2 (τ) σ

1

2 log R(D τ)

d2

(m − m

c

)

that is R(τ) = R 0 (τ) ≤ R 0 (τ) where R 1 dR

0

= R 0 2 (τ ) σ (0) −

d2

(m − m

c

)

and asymptotically as τ → ∞ , R(τ) = R 0 (τ − δ) for some delay δ > 0

(38)

B. Fast diffusion equations:

New points of view

1

improved inequalities and scalings

2

scalings and a concavity property

3

improved rates and best matching

(39)

Improved inequalities and scalings

(40)

The logarithmic Sobolev inequality

d µ = µ dx, µ(x) = (2 π) −d/2 e −|x|

2

/2 , on R d with d ≥ 1 Gaussian logarithmic Sobolev inequality

Z

R

2

|∇ u | 2 dµ ≥ 1 2

Z

R

2

| u | 2 log | u | 2 dµ for any function u ∈ H 1 ( R d , dµ) such that R

R

2

| u | 2 dµ = 1 ϕ(t ) := d

4

exp 2 t

d

− 1 − 2 t d

∀ t ∈ R

[Bakry, Ledoux (2006)], [Fathi et al. (2014)], [Dolbeault, Toscani (2014)]

Proposition

Z

R

2

|∇ u | 2 dµ − 1 2

Z

R

2

| u | 2 log | u | 2 dµ ≥ ϕ Z

R

2

| u | 2 log | u | 2

∀ u ∈ H 1 ( R d , d µ) s.t.

Z

R

2

| u | 2 d µ = 1 and Z

R

2

| x | 2 | u | 2 dµ = d

(41)

Consequences for the heat equation

Ornstein-Uhlenbeck equation (or backward Kolmogorov equation)

∂f

∂t = ∆f − x · ∇ f

with initial datum f 0 ∈ L 1 + ( R d , (1 + | x | 2 ) d µ and define the entropy as E [f ] :=

Z

R

2

f log f d µ , d

dt E [f ] = − 4 Z

R

2

|∇ √

f | 2 dµ ≤ − 2 E [f ] thus proving that E [f (t, · )] ≤ E [f 0 ] e 2t . Moreover,

d dt

Z

R

2

f | x | 2 dµ = 2 Z

R

2

f (d − | x | 2 ) dµ Theorem

Assume that E [f 0 ] is finite and R

R

2

f 0 | x | 2 dµ = d R

R

2

f 0 dµ. Then E [f (t , · )] ≤ − d

2 log h 1 −

1 − e

2d

E [f

0

] e 2t i

∀ t ≥ 0

(42)

Gagliardo-Nirenberg inequalities and the FDE

k∇ w k ϑ L

2

( S

d

) k w k 1 L

q+1

ϑ ( S

d

) ≥ C GN k w k L

2q

( S

d

)

With the right choice of the constants, the functional J[w ] := 1

4 (q 2 − 1) Z

R

d

|∇ w | 2 dx+β Z

R

d

| w | q+1 dx −K C α GN

Z

R

d

| w | 2q dx

2qα

is nonnegative and J[w ] ≥ J[w ] = 0 Theorem

[Dolbeault-Toscani] For some nonnegative, convex, increasing ϕ J[w ] ≥ ϕ

β R

R

d

| w | q+1 dx − R

R

d

| w | q+1 dx for any w ∈ L q+1 ( R d ) such that R

R

d

|∇ w | 2 dx < ∞ and R

R

d

| w | 2q | x | 2 dx = R

R

d

w 2q | x | 2 dx

Consequence for decay rates of relative R´enyi entropies:

see [Carrillo-Toscani]

(43)

Scalings and a concavity property

(44)

The fast diffusion equation in original variables

Consider the nonlinear diffusion equation in R d , d ≥ 1

∂u

∂t = ∆u p

with initial datum u(x, t = 0) = u 0 (x) ≥ 0 such that R

R

d

u 0 dx = 1 and R

R

d

| x | 2 u 0 dx < + ∞ . The large time behavior of the solutions is governed by the source-type Barenblatt solutions

U (t, x) := 1

κ t 1/µ d B x κ t 1/µ

where

µ := 2 + d (p − 1) , κ := 2 µ p p − 1

1/µ

and B ⋆ is the Barenblatt profile B ⋆ (x) :=

 

C ⋆ − | x | 2 1/(p − 1)

+ if p > 1 C ⋆ + | x | 2 1/(p− 1)

if p < 1

(45)

The entropy

The entropy is defined by E :=

Z

R

d

u p dx and the Fisher information by

I :=

Z

R

d

u |∇ v | 2 dx with v = p p − 1 u p 1 If u solves the fast diffusion equation, then

E = (1 − p) I To compute I , we will use the fact that

∂v

∂t = (p − 1) v ∆v + |∇ v | 2 F := E σ with σ = µ

d (1 − p) = 1+ 2 1 − p

1

d + p − 1

= 2 d

1

1 − p − 1

has a linear growth asymptotically as t → + ∞

(46)

The concavity property

Theorem

[Toscani-Savar´e] Assume that p ≥ 1 − 1 d if d > 1 and p > 0 if d = 1.

Then F (t) is increasing, (1 − p) F ′′ (t ) ≤ 0 and

t→ lim + ∞

1

t F(t) = (1 − p) σ lim

t→ + ∞ E σ 1 I = (1 − p) σ E σ 1 I ⋆

[Dolbeault-Toscani] The inequality

E σ 1 I ≥ E σ 1 I ⋆

is equivalent to the Gagliardo-Nirenberg inequality k∇ w k θ L

2

( S

d

) k w k 1 L

q+1

θ ( S

d

) ≥ C GN k w k L

2q

( S

d

)

if 1 − 1 d ≤ p < 1. Hint: u p 1/2 = kwk w

L2q(Sd)

, q = 2 p− 1 1

(47)

The proof

Lemma

I = d dt

Z

R

d

u |∇ v | 2 dx = − 2 Z

R

d

u p

k D 2 v k 2 + (p − 1) (∆v) 2 dx

k D 2 v k 2 = 1

d (∆v) 2 +

D 2 v − 1 d ∆v Id

2

1

σ (1 − p) E 2 σ (E σ ) ′′ = (1 − p) (σ − 1) Z

R

d

u |∇ v | 2 dx 2

− 2 1

d + p − 1 Z

R

d

u p dx Z

R

d

u p (∆v ) 2 dx

− 2 Z

R

d

u p dx Z

R

d

u p

D 2 v − 1 d ∆v Id

2

dx

(48)

Improved rates and best matching

(49)

Temperature (fast diffusion case)

The second moment functional (temperature) is defined by Θ(t ) := 1

d Z

R

d

| x | 2 u(t, x) dx and such that

Θ = 2 E

0 t

τ(0)

τ(t) τ

= lim

s→+∞

τ (s) s Θ

µ/2

(s) s (s τ(0))Θ

s (s

+

τ(t))Θ

µ/2

µ/2

s (s

+

τ

µ/2

s

Fast diffusion case

+

(50)

Temperature (porous medium case) and delay

Let U ⋆ s be the best matching Barenblatt function, in the sense of relative entropy F [u | U ⋆ s ], among all Barenblatt functions ( U ⋆ s ) s>0 . We define s as a function of t and consider the delay given by

τ(t ) :=

Θ(t ) Θ ⋆

µ2

− t

0

s τ(t) τ(0)

τ

= lim

s→+∞

τ(s) t

s (s

+

τ

(0)) Θ

µ/2

s Θ

µ/2

(s)

s (s

+

τ (t))

Θ

µ/2

s (s

+

τ

) Θ

µ/2

Porous medium case

(51)

A result on delays

Theorem

Assume that p ≥ 1 − 1 d and p 6 = 1. The best matching Barenblatt function of a solution u is (t, x) 7→ U (t + τ(t), x) and the function t 7→ τ(t ) is nondecreasing if p > 1 and nonincreasing if 1 − d 1 ≤ p < 1 With G := Θ 1

η2

, η = d (1 − p) = 2 − µ, the R´enyi entropy power functional H := Θ

η2

E is such that

G = µ H with H := Θ

η2

E H

1 − p = Θ 1

η2

Θ I − d E 2

= d E 2

Θ

η2

+1 (q − 1) with q := Θ I d E 2 ≥ 1 d E 2 = 1

d

− Z

R

d

x · ∇ (u p ) dx 2

= 1 d

Z

R

d

x · u ∇ v dx 2

≤ 1 d

Z

R

d

u | x | 2 dx Z

R

d

u |∇ v | 2 dx = Θ I

(52)

An estimate of the delay

Theorem

If p > 1 − 1 d and p 6 = 1, then the delay satisfies

t → lim + ∞ | τ(t ) − τ(0) | ≥ | 1 − p | Θ(0) 1

d2

(1 −p) 2 H ⋆

H ⋆ − H(0) 2

Θ(0) I(0) − d E(0) 2

(53)

C. Fast diffusion equations on manifolds and sharp functional

inequalities

1

The sphere

2

The line

3

Compact Riemannian manifolds

4

The cylinder: Caffarelli-Kohn-Nirenberg

inequalities

(54)

Interpolation inequalities on the sphere

Joint work with M.J. Esteban, M. Kowalczyk and M. Loss

(55)

A family of interpolation inequalities on the sphere

The following interpolation inequality holds on the sphere p − 2

d Z

S

d

|∇ u | 2 d v g + Z

S

d

| u | 2 d v g ≥ Z

S

d

| u | p d v g

2/p

∀ u ∈ H 1 ( S d , dv g ) for any p ∈ (2, 2 ] with 2 = d 2d 2 if d ≥ 3

for any p ∈ (2, ∞ ) if d = 2

Here dv g is the uniform probability measure: v g ( S d ) = 1

1 is the optimal constant, equality achieved by constants

p = 2 corresponds to Sobolev’s inequality...

(56)

Stereographic projection

(57)

Sobolev inequality

The stereographic projection of S d ⊂ R d × R ∋ (ρ φ, z) onto R d : to ρ 2 + z 2 = 1, z ∈ [ − 1, 1], ρ ≥ 0, φ ∈ S d− 1 we associate x ∈ R d such that r = | x | , φ = | x x |

z = r 2 − 1

r 2 + 1 = 1 − 2

r 2 + 1 , ρ = 2 r r 2 + 1

and transform any function u on S d into a function v on R d using u(y ) = ρ r

d−22

v(x) = r

2

2 +1

d−22

v(x) = (1 − z )

d−22

v(x) p = 2 , S d = 1 4 d (d − 2) | S d | 2/d : Euclidean Sobolev inequality

Z

R

d

|∇ v | 2 dx ≥ S d

Z

R

d

| v |

d2−2d

dx

dd2

∀ v ∈ D 1,2 ( R d )

(58)

Extended inequality

Z

S

d

|∇ u | 2 d v g ≥ d p − 2

"Z

S

d

| u | p d v g

2/p

− Z

S

d

| u | 2 d v g

#

∀ u ∈ H 1 ( S d , dµ) is valid

for any p ∈ (1, 2) ∪ (2, ∞ ) if d = 1, 2 for any p ∈ (1, 2) ∪ (2, 2 ] if d ≥ 3 Case p = 2: Logarithmic Sobolev inequality Z

S

d

|∇ u | 2 d v g ≥ d 2

Z

S

d

| u | 2 log

R | u | 2

S

d

| u | 2 d v g

d v g ∀ u ∈ H 1 ( S d , dµ) Case p = 1: Poincar´e inequality

Z

S

d

|∇ u | 2 d v g ≥ d Z

S

d

| u − u ¯ | 2 d v g with u ¯ :=

Z

S

d

u d v g ∀ u ∈ H 1 ( S d , dµ)

(59)

Optimality: a perturbation argument

For any p ∈ (1, 2 ] if d ≥ 3, any p > 1 if d = 1 or 2, it is remarkable that

Q [u] := (p − 2) k∇ u k 2 L

2

( S

d

)

k u k 2 L

p

( S

d

) − k u k 2 L

2

( S

d

)

≥ inf

u ∈ H

1

( S

d

,dµ) Q [u] = 1 d is achieved in the limiting case

Q [1 + ε v] ∼ k∇ v k 2 L

2

( S

d

)

k v k 2 L

2

( S

d

)

as ε → 0 when v is an eigenfunction associated with the first nonzero eigenvalue of ∆ g , thus proving the optimality

p < 2: a proof by semi-groups using Nelson’s hypercontractivity lemma. p > 2: no simple proof based on spectral analysis is available:

[Beckner], an approach based on Lieb’s duality, the Funk-Hecke formula and some (non-trivial) computations

elliptic methods / Γ 2 formalism of Bakry-Emery / nonlinear flows

(60)

Schwarz symmetrization and the ultraspherical setting

(ξ 0 , ξ 1 , . . . ξ d ) ∈ S d , ξ d = z, P d

i=0 | ξ i | 2 = 1 [Smets-Willem]

Lemma

Up to a rotation, any minimizer of Q depends only on ξ d = z

• Let dσ(θ) := (sin Z θ)

dd−1

d θ, Z d := √ π Γ(

d 2 ) Γ( d+1 2 )

: ∀ v ∈ H 1 ([0, π], dσ) p − 2

d Z π

0 | v (θ) | 2 dσ + Z π

0 | v (θ) | 2 d σ ≥ Z π

0 | v (θ) | p

2p

• Change of variables z = cos θ, v(θ) = f (z ) p − 2

d Z 1

− 1 | f | 2 ν dν d + Z 1

− 1 | f | 2 dν d ≥ Z 1

− 1 | f | p dν d

2p

where ν d (z ) dz = dν d (z) := Z d 1 ν

d2

1 dz , ν(z ) := 1 − z 2

(61)

The ultraspherical operator

With dν d = Z d 1 ν

d2

1 dz, ν(z) := 1 − z 2 , consider the space L 2 (( − 1, 1), dν d ) with scalar product

h f 1 , f 2 i = Z 1

− 1

f 1 f 2 dν d , k f k p = Z 1

− 1

f p dν d

p1

The self-adjoint ultraspherical operator is

L f := (1 − z 2 ) f ′′ − d z f = ν f ′′ + d 2 ν f which satisfies h f 1 , L f 2 i = − R 1

− 1 f 1 f 2 ν dν d

Proposition

Let p ∈ [1, 2) ∪ (2, 2 ], d ≥ 1

−h f , L f i = Z 1

− 1 | f | 2 ν dν d ≥ d k f k 2 p − k f k 2 2

p − 2 ∀ f ∈ H 1 ([ − 1, 1], d ν d )

(62)

Flows on the sphere

Heat flow and the Bakry-Emery method

Fast diffusion (porous media) flow and the choice of the exponents

Joint work with M.J. Esteban, M. Kowalczyk and M. Loss

(63)

Heat flow and the Bakry-Emery method

With g = f p , i.e. f = g α with α = 1/p

(Ineq.) −h f , L f i = −h g α , L g α i =: I [g ] ≥ d k g k 2 1 α − k g 2 α k 1

p − 2 =: F [g ] Heat flow

∂g

∂t = L g d

dt k g k 1 = 0 , d

dt k g 2 α k 1 = − 2 (p − 2) h f , L f i = 2 (p − 2) Z 1

− 1 | f | 2 ν dν d

which finally gives d

dt F [g (t , · )] = − d p − 2

d

dt k g 2 α k 1 = − 2 d I [g (t, · )]

Ineq. ⇐⇒ d

dt F [g (t , · )] ≤ − 2 d F [g (t, · )] ⇐ = d

dt I [g (t, · )] ≤ − 2 d I [g (t , · )]

(64)

The equation for g = f p can be rewritten in terms of f as

∂f

∂t = L f + (p − 1) | f | 2 f ν

− 1 2

d dt

Z 1

− 1 | f | 2 ν dν d = 1 2

d

dt h f , L f i = hL f , L f i + (p − 1) h | f | 2 f ν, L f i d

dt I [g (t, · )] + 2 d I [g (t, · )] = d dt

Z 1

− 1 | f | 2 ν dν d + 2 d Z 1

− 1 | f | 2 ν dν d

= − 2 Z 1

− 1

| f ′′ | 2 + (p − 1) d d + 2

| f | 4

f 2 − 2 (p − 1) d − 1 d + 2

| f | 2 f ′′

f

ν 2 dν d

is nonpositive if

| f ′′ | 2 + (p − 1) d d + 2

| f | 4

f 2 − 2 (p − 1) d − 1 d + 2

| f | 2 f ′′

f is pointwise nonnegative, which is granted if

(p − 1) d − 1 d + 2

2

≤ (p − 1) d

d + 2 ⇐⇒ p ≤ 2 d 2 + 1

(d − 1) 2 = 2 # < 2 d

d − 2 = 2

(65)

... up to the critical exponent: a proof in two slides

d dz , L

u = ( L u) − L u = − 2 z u ′′ − d u Z 1

− 1

( L u) 2 dν d = Z 1

− 1 | u ′′ | 2 ν 2 d ν d + d Z 1

− 1 | u | 2 ν dν d

Z 1

− 1

( L u) | u | 2

u ν dν d = d d + 2

Z 1

− 1

| u | 4

u 2 ν 2 dν d − 2 d − 1 d + 2

Z 1

− 1

| u | 2 u ′′

u ν 2 dν d

On ( − 1, 1), let us consider the porous medium (fast diffusion) flow u t = u 2

L u + κ | u | 2 u ν

If κ = β (p − 2) + 1, the L p norm is conserved d

dt Z 1

− 1

u βp dν d = β p (κ − β (p − 2) − 1) Z 1

− 1

u β(p 2) | u | 2 ν dν d = 0

(66)

f = u β , k f k 2 L

2

( S

d

) + p d 2

k f k 2 L

2

( S

d

) − k f k 2 L

p

( S

d

)

≥ 0 ?

A :=

Z 1

− 1 | u ′′ | 2 ν 2 dν d − 2 d − 1

d + 2 (κ + β − 1) Z 1

− 1

u ′′ | u | 2 u ν 2 dν d

+

κ (β − 1) + d

d + 2 (κ + β − 1) Z 1

− 1

| u | 4 u 2 ν 2 dν d

A is nonnegative for some β if 8 d 2

(d + 2) 2 (p − 1) (2 − p) ≥ 0

A is a sum of squares if p ∈ (2, 2 ) for an arbitrary choice of β in a certain interval (depending on p and

A = Z 1

− 1

u ′′ − p + 2 6 − p

| u | 2 u

2

ν 2 dν d ≥ 0 if p = 2 and β = 4

6 − p

(67)

The rigidity point of view

Which computation have we done ? u t = u 2

L u + κ |u

|

2

u ν

− L u − (β − 1) | u | 2

u ν + λ

p − 2 u = λ p − 2 u κ Multiply by L u and integrate

...

Z 1

− 1 L u u κ dν d = − κ Z 1

− 1

u κ | u | 2 u dν d

Multiply by κ | u

|

2

u and integrate ... = + κ

Z 1

− 1

u κ | u | 2 u dν d

The two terms cancel and we are left only with the two-homogenous

terms

(68)

Improvements of the inequalities (subcritical range)

as long as the exponent is either in the range (1, 2) or in the range (2, 2 ), on can establish improved inequalities

An improvement automatically gives an explicit stability result of the optimal functions in the (non-improved) inequality

By duality, this provides a stability result for Keller-Lieb-Tirring

inequalities

(69)

What does “improvement” mean ?

An improved inequality is

d Φ (e) ≤ i ∀ u ∈ H 1 ( S d ) s.t. k u k 2 L

2

( S

d

) = 1 for some function Φ such that Φ(0) = 0, Φ (0) = 1, Φ > 0 and Φ(s) > s for any s. With Ψ(s) := s − Φ 1 (s)

i − d e ≥ d (Ψ ◦ Φ)(e) ∀ u ∈ H 1 ( S d ) s.t. k u k 2 L

2

( S

d

) = 1 Lemma (Generalized Csisz´ar-Kullback inequalities)

k∇ u k 2 L

2

( S

d

) − d p − 2

h k u k 2 L

p

( S

d

) − k u k 2 L

2

( S

d

)

i

≥ d k u k 2 L

2

( S

d

) (Ψ ◦ Φ)

C k u k

2 (1−r) Ls(Sd)

k u k

2L2 (Sd)

k u r − u ¯ r k 2 L

q

( S

d

)

∀ u ∈ H 1 ( S d ) s(p) := max { 2, p } and p ∈ (1, 2): q(p) := 2/p, r (p) := p; p ∈ (2, 4):

q = p/2, r = 2; p ≥ 4: q = p/(p − 2), r = p − 2

(70)

Linear flow: improved Bakry-Emery method

Cf. [Arnold, JD]

w t = L w + κ | w | 2 w ν With 2 := (d− 2 d

2

+1 1)

2

γ 1 :=

d − 1 d + 2

2

(p − 1) (2 # − p) if d > 1 , γ 1 := p − 1

3 if d = 1 If p ∈ [1, 2) ∪ (2, 2 ] and w is a solution, then

d

dt (i − d e) ≤ − γ 1

Z 1

− 1

| w | 4

w 2 dν d ≤ − γ 1 | e | 2 1 − (p − 2) e Recalling that e = − i, we get a differential inequality

e ′′ + d e ≥ γ 1 | e | 2

1 − (p − 2) e

After integration: d Φ(e(0)) ≤ i(0)

(71)

Nonlinear flow: the H¨older estimate of J. Demange

w t = w 2

L w + κ | w | 2 w

For all p ∈ [1, 2 ], κ = β (p − 2) + 1, dt d R 1

− 1 w βp dν d = 0

1

2

dt d R 1

− 1

| (w β ) | 2 ν + p d 2 w − w

dν d ≥ γ R 1

− 1

|w

|

4

w

2

ν 2 dν d

Lemma

For all w ∈ H 1 ( − 1, 1), dν d

, such that R 1

− 1 w βp dν d = 1 Z 1

− 1

| w | 4

w 2 ν 2 dν d ≥ 1 β 2

R 1

− 1 | (w β ) | 2 ν d ν d R 1

− 1 | w | 2 ν dν d

R 1

− 1 w dν d

δ

.... but there are conditions on β

(72)

Admissible (p, β ) for d = 5

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

1

2

3

4

5

6

(73)

The line

A first example of a non-compact manifold

Joint work with M.J. Esteban, A. Laptev and M. Loss

(74)

One-dimensional Gagliardo-Nirenberg-Sobolev inequalities

k f k L

p

( R ) ≤ C GN (p) k f k θ L

2

( R ) k f k 1 L

2

( θ R ) if p ∈ (2, ∞ ) k f k L

2

( R ) ≤ C GN (p) k f k η L

2

( R ) k f k 1 L

p

( η R ) if p ∈ (1, 2) with θ = p 2 p 2 and η = 2 2+p p

The threshold case corresponding to the limit as p → 2 is the logarithmic Sobolev inequality

R

R u 2 log

u

2

k u k

2L2 (R)

dx ≤ 1 2 k u k 2 L

2

( R ) log

2 π e

ku

k

2L2 (R)

k u k

2L2 (R)

If p > 2, u ⋆ (x) = (cosh x)

p22

solves

− (p − 2) 2 u ′′ + 4 u − 2 p | u | p− 2 u = 0

If p ∈ (1, 2) consider u (x) = (cos x)

22p

, x ∈ ( − π/2, π/2)

(75)

Flow

Let us define on H 1 ( R ) the functional F [v] := k v k 2 L

2

( R ) + 4

(p − 2) 2 k v k 2 L

2

( R ) − C k v k 2 L

p

( R ) s.t. F [u ⋆ ] = 0 With z (x) := tanh x , consider the flow

v t = v 1

p2

√ 1 − z 2

v ′′ + 2 p

p − 2 z v + p 2

| v | 2

v + 2

p − 2 v

Theorem (Dolbeault-Esteban-Laptev-Loss) Let p ∈ (2, ∞ ). Then

d

dt F [v(t)] ≤ 0 and lim

t →∞ F [v(t)] = 0

d

dt F [v (t )] = 0 ⇐⇒ v 0 (x) = u ⋆ (x − x 0 )

Similar results for p ∈ (1, 2)

Références

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